Let's recycle sequences.
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If I've got a sequence in hand, I can
build a new sequence out of that old sequence.
Well, here's a sequence to start with.
It's a sequence, the first three terms of which, are all 1.
So a sub 0, a sub 1, a sub 2 are all equal to 1.
And then to compute each subsequent term I just add up the three previous terms.
Let's see some terms.
Well, here are some terms in this sequence.
The first three terms are all 1. 1, 1, 1, a sub zero, a sub 1, a sub 2.
Each subsequent term is computed using this recursive formula.
What this formula tells me to do is to add up the previous three terms.
So this 3 is coming from 1 plus 1 plus 1. This 5 is coming from 3 plus 1 plus 1.
This 9 is coming from 5 plus 3 plus 1. This 17 is coming from 9 plus 5 plus
3, and so on.
Now this might seem a little bit like the Fibonacci sequence, except instead
of adding up the previous 2 terms, I'm adding up the previous 3 terms.
So, to be a little bit funny,
people sometimes call this the Tribonacci sequence.
I can build a new sequence out of the Tribonacci sequence.
So starting with this sequence, I could build a new sequence,
sequence I'll call b, by referring to the terms in this
Tribonacci sequence. Let's see some terms of this sequence b.
Here is the sequence a sub n.
All right. It goes 1, 1, 1, 3, 5.
This is the Tribonacci sequence.
And here is the beginning of the sequence b sub n.
According to this rule, b sub n is just
the ratio of neighboring terms in the tribonacci sequence.
So, for example, this term here,
which is b sub 0, b sub 1, b sub 2, b sub 3, is the ratio of a sub 4
and a sub 3. Or this term here, which is b sub 4, is
the ratio of a sub 5 and a sub 4. The fractions are sort of confusing.
So we can replace these fractions with decimal approximations.
Instead of looking at these fractions.
Here are some of the terms of b sub n,
but written out in terms of their decimal approximations.
Let's look at these approximations.
Remember, these are decimal approximations to
neighboring terms in the Tribonacci sequence.
And as we go out further and further in the Tribonacci sequence.
Do you notice something?
Well, in light of this numeric evidence, it
certainly looks like this sequence, b sub n,
has a limit. And this limit is about 1.839.
So that raises a question.
So that's a puzzle for you. Yeah, this limit turns out to exist.
And it's about 1.8.
But what is it exactly? What is this limit exactly equal to?
Can you find a, a formula for this limit?
That turns out to be an extremely challenging question.
But I hope it wets your appetite.
With not
very many tools at hand, it's possible to build relatively
simple seeming things that are actually quite complicated to describe precisely.
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